A Relation between Standard Conjectures and Their Arithmetic Analogues

Ap(X,H) is called the hard Lefschetz conjecture and Hp(X,H) is called the Hodge index conjecture. When the characteristic of k is zero, the Hodge index conjecture is already proved. On the other hand, for an arithmetic variety the intersection theory of cycles was established by Arakelov [1] for surfaces and Gillet and Soulé [5] for higher dimensional varieties. It is quite natural to ask whether analogues of standard conjectures hold in this situation. We now explain this. Let X be a regular scheme which is projective and flat over Z. We assume that the generic fiber XQ is smooth over Q. Such a scheme is called an arithmetic variety. For an arithmetic variety X the arithmetic Chow group ĈH p (X) is defined and the intersection product on